Oriented Mixed Area and Discrete Minimal Surfaces

Abstract. Recently a curvature theory for polyhedral surfaces has been established which associates with each face a mean curvature value computed from areas and mixed areas of that face and its corresponding Gauss image face. Therefore a study of minimal surfaces requires studying pairs of polygons with vanishing mixed area. We show that the mixed area of two edgewise parallel polygons equals the mixed area of a derived polygon pair which has only the half number of vertices. Thus we are able to recursively characterize vanishing mixed area for hexagons and other n-gons in an incidence-geometric way. We use these geometric results for the construction of discrete minimal surfaces and a study of equilibrium forces in their edges, especially those with the combinatorics of a hexagonal mesh. Contents

• Type: Journal
• Year: 2010
• Where: DCG
• Authors: Christian Müller, Johannes Wallner
• Keywords: DCG 2010, Edgewise Parallel Polygons, Mean Curvature Value, Mixed Area